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International Journal of Engineering and Advanced Technology (IJEAT)
ISSN: 2249 8958, Volume-2, Issue-4, April 2013
759
Transient Stability of a Multi Machine Power
SystemDevender Kumar, Balwinder Singh Surjan
Abstract -Transient stabil ity analysis has recentl y become a
major issue in the operation of power systems due to the
increasing stress on power system networks. This problem
requires evaluation of a power system's ability to withstand
disturbances while maintaining the quality of service. Many
dif ferent techni ques have been proposed for tr ansient stabil ity
analysis in power systems, special ly f or a multimachine system.
These methods include the time domain solu tions, the extended
equal area cri teria, and the direct stabil ity methods such as the
transient energy function. H owever, the most methods must
transform f rom a multi -machine system to an equi valent machine
and in fi ni te bus system [1] [3] .Thi s paper i ntr oduces a method as
an accurate algorithm to analyse transient stability for power
system with an individual machine. It i s as a tool to identif ystable and unstable conditions of a power system after fault
clearing with solving dif ferential equations. [5][6] .
Key words- multimachine power system, matlab Simulink,
transient stabil ity, damping
I. INTRODUCTIONMultimachine equations can be written Similar to the one-
machine system connected to the infinite bus. In order to
reduce the complexity of the transient stability analysis,
similar simplifying assumptions are made as follows. -Eachsynchronous machine is represented by a constant voltage
source behind the direct axis transient reactance. Thisrepresentation Neglects the effect of saliency and assumes
Constant flux linkages. The governors action are neglected
and the input powers are assumed to remain constant during
the entire period of simulation [4].
-Using the prefault bus voltages, all loads are converted to
equivalent admittances to ground and are assumed to remain
constant.
-Damping or asynchronous powers are
Ignored. The mechanical rotor angle of each machinecoincides with the angle of the voltage behind the machine
reactance.
-Machines belonging to the same station swing together and
are said to be coherent. A group of coherent machines isrepresented by one equivalent machine
II. MATHEMATICAL MODEL OF MULTIMACHINETRANSIENT STABILITY ANALYSIS
The first step in the transient stability analysis is to solve the
initial load flow and to determine the initial bus voltage
magnitudes and phase angles. The machine currents prior to
disturbance are calculated from[5].
Manuscript received on April, 2013.
Devender kumar, Post graduate student at PEC University of
technology, Chandigarh, India.Dr. Balwinder singh surjan, Associate Proffesor and Head of the
department of PEC University of technology, Chandigarh, India.
Ii= Si/Vi*= (PijQi)/Vi
*, i= 1,2,3m (1)
Where
m is the number of generators
Viis the terminal voltage of the ith generator
Piand Qi are the generator real and reactive
powers.
All unknow values are determined from the initial power
flow solution. The generator armature resistances are
usually neglected and the voltages behind the transient
reactances are then obtained[6]
Ei=Vi+jXdIi (2)
Next, all load are converted to equivalent
admittances by using the relation
Yio= Si*/Vi
2= (Pi-jQi)/Vi
2 (3)
To include voltages behind transientreactances, m buses are added to the n bus
power system network.
Fig 1 Power system representation for multi machine
stability studies
In this system one generator is taken as reference generator
and other two are studied for stability purposes. Fault occur
at point p in the system, and two loads are connected to thesystem at Sd1and Sd2
Ibus= YbusVbus (4)
Where
Ibus is the vector of the injected bus currents
Vbus is the vector of bus voltages measured
from the reference node
Prefault bus matrixYprefault= Yl4+ Y41+ Y45 Y24+B41/2+B45/2 (5)
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Transient Stability of a Multi Machine Power System
760
Yprefault = Yl5+Y54+ Y51+ Y35+ B54/2 + B51/2
(6)
B--- charging reactance of the system
During fault bus matrix
Since the fault is near the bus, so it is short circuited to
ground.
Ybus= YjoldYnoldYnjold/ Ynnold (7)
Post fault bus matrixOnce the fault is cleared by removing the line,
simultaneously opening the circuit breaker at the either ends
of the line between buses, prefault Ybushas to be modified
again.
Ypostfault= YjjoldYji - Bij/2 (8)
The diagonal elements of the bus admittance
matrix are the sum of admittances connected to it, and the
off-diagonal elements are equal to the negative of the
admittance between the nodes. The reference is that
additional nodes are added to include the machine voltages
behind transient reactances. Also, diagonal elements aremodified to include the load admittances. To simplify the
analysis, all nodes other than the generator internal nodes
are eliminated using Kron reduction formula [5]. To
eliminate the load buses, the bus admittance matrix in (4) ispartitioned such that the n buses to be removed are
represented in the upper n rows. Since no current enters or
leaves the load buses, currents in the n rows are zero. The
generator currents are denoted by the vector Im and the
generator and load voltages are represented by the vector E
m and Vn, respectively..
During fault power angle equation
Pe2= 0 (9)
Pe3= Re[ Y33E3E
*+ E3
* Y31E1
.],
since Y32=0
= (E3)
2G33+ E1
E3
Y31cos (3131)
(10)
Post fault power angle equations
Pe2= E22G22+ E1E2Y21cos(2121)
(11)
Pe3= E32G33+ E1E3Y31cos (3131)
(12)
Swing equations during fault
D22/dt
2= 180f/H2(Pm2Pe2) = 180f/H2Pa2
(13)
D23/dt
2= 180f/H3(Pm3Pe3) (14)
Swing equation post fault
D22/dt
2= 180f/11[3.25- {0.6012 +8.365 sin (21.662
o)}]
(15)
D23/dt
2= 180f/9 [2.10-{0.1823+6.5282 sin (30.8466
o)}]
(16)
Pa= Pm - PcPmaxsin( ) (17)
The above swing equations can be solved by point to point
method
The classical transient stability study is based on the
application of a three-phase fault. A solid three-phase fault
at bus k in the network results in Vk= 0. This is simulatedby removing the kth row and column from the prefault bus
admittance matrix. The new bus admittance matrix is
reduced by eliminating all nodes except the internal
generator nodes. The generator excitation voltages during
the fault and postfault modes are assumed to remain
constant.In transient stability analysis problem, we have two state
equations for each generator. When the fault is cleared,
which may involve the removal of the faulty line, the bus
admittance matrix is recomputed to reflect the change in the
networks. Next the postfault reduced bus admittance matrix
is evaluated and the postfault electrical power of the ith
generator.
III. SIMULATIONBy using all the mathematical equations, the Simulink
diagram for multimachine stability is generated. The
Simulink diagram is highly complicated so it is divided intosubsystem 1 and subsystem 2.
Fig 2 subsystem 1 of multimachine power system
Fig 3 Subsystem 2 generated for multimachine power
system
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International Journal of Engineering and Advanced Technology (IJEAT)
ISSN: 2249 8958, Volume-2, Issue-4, April 2013
761
From fig 2, Pe2is generated and Pe3is generated from fig 3,
Pe2is the electromechanical power for machine 2, and P e3is
the electromechanical power for machine 3.Electromechanical power is generated by using voltage and
current of the machine and reactances of pre fault and post
fault condition. A switch is used for switching between pre
fault and post condition of the system. By using the outputs
of fig 2 and fig 3 multimachine system is generated. From
the prefault load flow data determine E voltage behindtransient reactance for all generators. This establishes
generator emf magnitudes which remain constant during the
study and initial rotor angle . Also record prime mover
inputs to generators, Pmk = Pgk.. Augment the load flow
network by the generator transient reactances. Shift network
buses behind transient reactances.
For faulted mode, find generator outputs from power angle
equations and solve swing equations step by step. Keep
repeating the the above step for post fault mode and after
line reclosure mode. Examine plots of all generator and
establish the answer to the stability question.
Fig 4 multimachine power system
The output of the multimachine power system is obtained
between torque angle v/s time. This is also known as swing
curve of the system. The output curves can be varied by
changing the critical clearing time if the system. When
critical clearing time of the system is low then both machinewould operate in stable operation, as the critical clearing
time of the system is increased then our system would move
towards instability. The machine which have more
oscillation would be more unstable as compared to the
machine which have less oscillation. The outputs are taken
at critical clearing time of 0.275 sec and 0.08 sec[7].
Fig 5 output response with critical clearing time 0.275 sec
Fig 6 output response with critical clearing time 0.08 sec
From the result it is seen that machine 1 is having less
oscillation than machine 2 at critical clearing time 0.08sec,
but both machine are stable and when critical clearing timeis increased to 0.275sec then machine 1 is still in stable
condition and machine 2 is in unstable condition.
IV. INTRODUCING DAMPING INTO THE SYSTEMDamping of the system is done to reduce the oscillation
present in the system. It is done by connecting a negative
gain of very low magnitude between speed and inertia gain
of the system. The gain which is used to damp out the
oscillation is known as the damping gain[8]. In this multi
machine system our two machine are present so we have toproduce damping in both these machine. Now we take three
cases of damping in multimachine system, these are asfollows.
Case 1when damping is done only in machine 1
Fig 7 output response when damping is introduced in
machine 1 only
By the above result we can see that our machine 1 output
become stable as it reaches a sable point and all the
oscillation are damp out and even the oscillation of machine
2 are decreased.
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Transient Stability of a Multi Machine Power System
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Case 2When damping is done only in machine 2
Fig 8 output response when damping is done only in
machine 2
By this response we can see that machine 2 is in stablecondition and machine 1 oscillation are increased
Case 3 when damping is done in both machine 1 and
machine 2
Fig 9 output response when damping is done in bath
machine 1 and machine 2
Now we can conclude that both machine 1 and machine 2
are in stable mode as both saturate at a point and oscillations
are removed from the system, so by introducing damping
into the system our system can be made stable from unstable
condition.
V. CONCLUSIONThis analysis allows to assess that the system is stable,
unstable and also allows to determine the critical clearingtime of power system with three-phase faults. These results
can be used effectively in planning or operation of power
systems.
REFERENCES
[1]. La Van Ut: Analysis and Control Stability for Power System,
Science and Technology Press, 2000.
[2]. Ho Van Hien: Transmission and Distribution of Power System,
National University of HoChiMinh City Press, 2003.[3]. Prabha Kundur: Power System Stability and Control, McGraw-Hill
International Editions, 1994.
[4]. Hadi Saadat: Power System Analysis, McGraw-Hill InternationalEditions, 1999.
[5]. John J. Grainer, Willam D. Stevenson JR: Power System Analysis,
McGraw-Hill International Editions, 1999.[6]. Carson W. Taylor: Power System Voltage Stability, McGraw-Hill
International Editions,1994.
[7]. I. J. Nagrath and D. P. Kothari Modern Power System AnalysisTata McGraw-Hill, 2003
[8] Power system stability by Mrinal K Pal